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4 BASES. DIMENSION AND CODIMENSION 369

Let V be a supplementary subspace of M in E. Then V is finite dimensional.
Let p: E-» V be the projection of E onto V with kernel M (A.3.4). The
subspace p(N) is finite dimensional (A.4.10). Let (b_i)₁^_i^_m be a basis of
XN), and for each z let c_teN be such that p(c_t) = 6_r. Then N is gener-
ated by M n N and the c_t (A.4.3), hence M n N is of finite codimension
in N and therefore of finite codimension in E. Let P (resp. Q) be a supple-
mentary subspace of M n N in M (resp. N), and let R be a supplementary
subspace of M + N in E. Then E is the direct sum of M n N, P, Q, and
R, so that we have codim(M n N) = dim P + dim Q + dim R, codim M =
dim Q + dim R, codim N = dim P 4- dim R, and codim(M -f N) = dim R.
The result (A.4.13.1) follows immediately from these formulas.

(A.4.14) A subspace of codimension 1 in a vector space E is called a hyper-
plane in E. If E is of finite dimension n, every hyperplane in E is of dimension
n-l (A.4.10.1).

(A.4.15) If His a hyperplane in E, there exists a linear form f=£ 0 on E such
thatf~"*(0) = H. Iff is another linear form on E such thatf'~^l(G) = H, then
there exists a scalar y ^ 0 such that f = yf. Conversely, if g is any nonzero
linear form on E, then g~^l(fy is a hyperplane in E.

If H is a hyperplane, there exists a vector a $ H such that E is the direct
sum of H and Ka, and every x e E can therefore be expressed uniquely in
the form x = y +f(x)a, where/(x) e K. Since x^f(x)a is linear (A.3.4), so
is x-+f(x) (A.2.3); hence/is a linear form, and H is its kernel (A.3.4). If
/' is any linear form such that /'"*(()) = H, and if we put f(a) = a and
f'(a) = /?, then we have a ^ 0 and /? ^ 0, and a/' — (If is a linear form on
E which vanishes on H and at a, and therefore vanishes identically on
E = H + Ka, This shows that /' = a'¹/?/ Finally, if H^/=^~¹(0), then
there exists a vector b $ H', because g ^ 0. Let y = g(b) ^ 0. Then for every
xeE we have g(x - y~V(x)£) = °> so that x=*y + y~^lg(x}b for some
y e H'. This shows that E = Kb H- H', and the sum is direct because b # H'.
Hence H' is a hyperplane.

(A.4.16) Let u : E -> F be a linear mapping. We say that u is of finite rank
if «(E) is of finite dimension. The dimension of u(E) is then called the rank
of u, and is written rank(w). If u(E) is infinite dimensional, then u is said to
be of infinite rank.